I think there is a clear difference between SteamOS and Stadia. Whilst SteamOS was only accessable by either buying a specific device or wiping your OS and installing a new fresh one. Stadia is cloud based and can be ran off phones, computers and practically anything that could be considered a reasonable gaming device. That's one advantage Stadia already has.
Furthermore, it has got some big developers like Bethesda and CD Projekt Red to release games like Doom Eternal and Cyberpunk 2077 respectively on Stadia. I do think Stadia does have a chance to take off it just needs to play it's cards right (and apart from the games being separate fiasco they are playing them well).
And yeah, Linux is a really small minority when it comes to PC Gaming in general. But I think that could change in a few years. Proton is already making massive progress to make Windows games run effortlessly using the Steam port for Linux. Also, Linux is getting more and more attention as an alternative due to a number of things like: driver support gradually getting better and better, proton maturing and Linux even outperforming Windows on certain titles.
I personally do think the future of gaming is likely going to be on Linux. Simply because it's getting more attention (thanks to Stadia and Proton), Linux having benefits that Windows does not have and it's becoming easier and easier to set it all up.
It's far from ready yet and still has a long road ahead until it can be considered a viable alternative for most Windows games. But it is getting there and it's making more progress by the day.
My point is that Linux based services and platforms do not magically garner more end-user Linux support. SteamOS was supposed to be the future of gaming, had the same "big dev" promises, was "proven" to be soooo much better than Windows for games...and it went absolutely nowhere. Why? Because the average person doesn't give a rats ass about that fancy Linux stuff, and couldn't be bothered to make the swap when they had something that already worked just fine. Stadia may have the advantage of being usable anywhere (so long as you meet the necessary ISP requirements), but cloud gaming still has plenty of hurdles it has to jump before it's even remotely considered a "viable alternative" to just having a PC or a console.
Also, Proton is just Wine for idiots with some additional tweaks that could've already been made by end users. If that were going to somehow magically increase Linux usage for the average consumer, it would've done so 20+ years ago when it launched, or 10+ years ago when PlayOnLinux (also Wine for idiots, but shit now unfortunately) launched. I fail to see why Wine for idiots would somehow make devs
more interested in releasing Linux ports when they'll see people are already getting games to run on Linux without them having to do anything. If anything, Proton simply proves to devs they don't need to even bother making a native Linux port, some dude will just make some tweaks using wine-config and winetricks and get it running well enough for those 20 other dudes who bother gaming on Linux.
As I said, Stadia does
nothing to promote the usage of Linux to the mass market (read: idiots), which is the real problem with devs not bothering to port anything to Linux. You have to remember that we're talking about people who are mostly not all that technologically inclined, you can spout your "DRIVER IMPROVEMENTS" and "MARGINALLY BETTER PERFORMANCE" such all you like, but none of that means jack shit to the majority of people who've gamed on Windows all their life and want something that "just works" that they're familiar with.
I love Linux and all, I dual boot it on pretty much every PC in my house and have occasionally gamed on it when I couldn't be bothered to reboot to Windows, but get real kid, it does not and will never have the necessary mass market appeal to be the "future of gaming" unless Microsoft magically goes bust and it suddenly becomes impossible to use Windows on any PC ever.
To put it more into perspective, macOS, the "LOL MAC HAS NO GAMES WHAT IDIOT WOULD GAME ON A MAC" platform has
3x more active Steam users per month than Linux has. And again, Linux only accounts for
0.79% of Steam users. There are 90 million active Steam accounts,
86.4 million are using Windows. If you honestly think Linux will
ever hit even 10% of that number in the next few decades, I'm afraid you're simply deluded.